Higher-order asymptotic crack-tip fields in simplified strain gradient elasticity

Abstract

Explicit representation for the higher-order in-plane crack-tip fields is derived using Papkovich–Neuber stress functions within the framework of simplified strain gradient elasticity (SGE). In the considered theory, strain energy density depends on both the infinitesimal strain tensor and its spatial gradient. The constitutive equations of simplified SGE contain an additional length scale parameter that can be correlated with the characteristic size of the material microstructure. Presented asymptotic solution has separable form and reduces to classical Williams’ series when the gradient effects are negligible. The leading terms in the derived solution coincide with the previously known asymptotic solutions for crack problems in SGE. The higher-order terms have coupled amplitude factors and modified definitions for the angular distribution as compared to the classical solution. Derived asymptotic fields are compared to the full-field numerical solution for the Mode I crack problem to quantify the amplitude factors and the zones of dominance for up to eight terms. The presented solution has practical applications in the development of advanced numerical methods, particularly concerning enriched finite elements for crack problems in SGE.